If `alpha, beta` are the roots of `x^(2) + x + 3=0` then `5alpha+alpha^(4)+ alpha^(3)+ 3alpha^(2)+ 5beta+3=`

Options

A. 0
B. -1
C. -2
D. -3

Answer

To solve the problem, we need to find the value of the expression \(5\alpha + \alpha^4 + \alpha^3 + 3\alpha^2 + 5\beta + 3\), given that \(\alpha\) and \(\beta\) are the roots of the quadratic equation \(x^2 + x + 3 = 0\).

### Step-by-Step Solution:

1. **Identify the roots**:
The roots of the equation \(x^2 + x + 3 = 0\) can be identified using Vieta's formulas:
- The sum of the roots \(\alpha + \beta = -\frac{b}{a} = -\frac{1}{1} = -1\).
- The product of the roots \(\alpha \beta = \frac{c}{a} = \frac{3}{1} = 3\).

2. **Rewrite the expression**:
We need to find the value of:
\[ 5\alpha + \alpha^4 + \alpha^3 + 3\alpha^2 + 5\beta + 3 \]
We can group the terms involving \(\alpha\) and \(\beta\):
\[ = 5(\alpha + \beta) + \alpha^4 + \alpha^3 + 3\alpha^2 + 3 \]

3. **Substitute \(\alpha + \beta\)**:
Substitute \(\alpha + \beta = -1\):
\[ = 5(-1) + \alpha^4 + \alpha^3 + 3\alpha^2 + 3 \]
\[ = -5 + \alpha^4 + \alpha^3 + 3\alpha^2 + 3 \]
\[ = \alpha^4 + \alpha^3 + 3\alpha^2 - 2 \]

4. **Use the original equation**:
Since \(\alpha\) is a root of the equation \(x^2 + x + 3 = 0\), we have:
\[ \alpha^2 + \alpha + 3 = 0 \implies \alpha^2 = -\alpha - 3 \]

5. **Calculate \(\alpha^3\) and \(\alpha^4\)**:
- To find \(\alpha^3\):
\[ \alpha^3 = \alpha \cdot \alpha^2 = \alpha(-\alpha - 3) = -\alpha^2 - 3\alpha \]
Substitute \(\alpha^2 = -\alpha - 3\):
\[ \alpha^3 = -(-\alpha - 3) - 3\alpha = \alpha + 3 - 3\alpha = -2\alpha + 3 \]

- To find \(\alpha^4\):
\[ \alpha^4 = \alpha \cdot \alpha^3 = \alpha(-2\alpha + 3) = -2\alpha^2 + 3\alpha \]
Substitute \(\alpha^2 = -\alpha - 3\):
\[ \alpha^4 = -2(-\alpha - 3) + 3\alpha = 2\alpha + 6 + 3\alpha = 5\alpha + 6 \]

6. **Substitute back into the expression**:
Now substitute \(\alpha^3\) and \(\alpha^4\) into the expression:
\[ = (5\alpha + 6) + (-2\alpha + 3) + 3(-\alpha - 3) - 2 \]
Simplifying this:
\[ = 5\alpha + 6 - 2\alpha + 3 - 3\alpha - 9 - 2 \]
\[ = (5\alpha - 2\alpha - 3\alpha) + (6 + 3 - 9 - 2) \]
\[ = 0\alpha + (-2) = -2 \]

### Final Answer:
Thus, the value of the expression \(5\alpha + \alpha^4 + \alpha^3 + 3\alpha^2 + 5\beta + 3\) is \(-2\).